Co a ian Mul i-Channel Windowed Sca e ing:
Phase–Densi y Uni ied Theo em
Au ic (S-se ies / EBOC)
Ve sion 1.2
No embe 24, 2025
Abs ac
Fo mul i-channel sca e ing wi h N(E) channels a ene gy E, es ablish uni ied
amewo k connec ing sca e ing phase de i a i e, ela i e spec al densi y, and Wigne –
Smi h delay. Co e o mula holding a.e. on absolu ely con inuous spec um:
1
N(E)
d
dE A g de S(E) = 1
2π⟨ Q(E)⟩N=ρ el(E)
whe e Q(E) = −iS†(E)∂ES(E) is Wigne –Smi h delay ma ix, ⟨·⟩Ndeno es pe -
channel a e age, ρ el ela i e spec al densi y om Bi man–K e˘ın o mula de S=
e−2πiξ.
Es ablish: (i) Channel co a iance: o mula in a ian unde uni a y channel ba-
sis ans o ma ions; (ii) Th eshold egula i y: handle channel numbe jumps ia
p ope bounda y condi ions; (iii) Windowed eadou : ini e-bandwid h measu emen
p o ocol wi h NPE e o closu e; (i ) In o ma ion geome y: Bo n p obabili y as
I-p ojec ion, poin e basis as Ky Fan minimum.
1 Se up and No a ion
Ene gy-dependen channel numbe N(E) wi h h eshold se T={E:N(E+)=N(E−)}.
Be ween h esholds, sca e ing ma ix S(E)∈U(N(E)) uni a y, di e en iable a.e.
Bi man–K e˘ın con en ion: de S(E) = e−2πiξ(E)whe e ξspec al shi unc ion, ρ el(E) =
−ξ′(E) ela i e spec al densi y.
Wigne –Smi h delay:Q(E) = −iS†(E)∂ES(E) sel -adjoin , eigen alues τj(E) indi id-
ual channel delays.
2 Main Resul s
Theo em 2.1 (Mul i-Channel Phase–Densi y Uni ica ion).On h eshold- egula in e als I
(whe e I∩ T =∅), ha e a.e.:
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2π Q(E) = d
dE A g de S(E) = −2π ξ′(E) = ρ el(E).
Pe -channel a e age delay ⟨τ⟩N(E) = 1
N(E) Q(E)sa is ies
⟨τ⟩N(E) = 2πℏρ el(E) ( es o ing ℏ).
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P oo . F om de S=e−2πiξ ge ∂Elog de S=−2πi ξ′. By Jacobi o mula ∂Elog de S=
(S−1∂ES) = (S†∂ES) (using uni a i y). Thus (S†∂ES) = −2πi ξ′, gi ing Q=−i (S†∂ES) =
2π ξ′. Bi man–K e˘ın ρ el =−ξ′comple es chain.
Theo em 2.2 (Channel Basis Co a iance).Unde uni a y channel ans o ma ion U∈
U(N),S7→ ˜
S=USU†, ha e
Q˜
S(E) = QS(E), ρ el[˜
S](E) = ρ el[S](E).
Windowed eadou Nw[S;E0] = Rw(E−E0)ρ el[S](E)dE in a ian .
P oo . T ace in a ian unde simila i y. Spec al shi unc ion gauge-in a ian .
Theo em 2.3 (Th eshold Bounda y Condi ions).A h eshold E∗∈ T whe e N(E−
∗) = N−,
N(E+
∗) = N+wi h N+> N−, impose:
1. Phase con inui y:A g de S(E)con inuous a E∗a e p ope b anch choice
2. Densi y egula iza ion:ρ el(E)may ha e δ- unc ion con ibu ion a E∗ om bound
s a es en e ing/lea ing con inuous spec um (Le inson heo em)
3. Windowed measu emen : choose window wwi h w(E∗−E0)su icien ly small o
smoo h o egula ize h eshold singula i ies
3 Windowed Readou and NPE E o
Theo em 3.1 (NPE Th ee-Te m Decomposi ion o Mul i-Channel).Fo windowed eadou
wi h window w, ke nel h, sampling s ep ∆E, unca ion N:
Nw[S;E0] = Zw(E−E0)[h∗ρ el](E)dE
disc e e app oxima ion
b
N= ∆E
N
X
n=−N
w(En−E0)[h∗ρ el](En)
sa is ies e o decomposi ion
|Nw−b
N| ≤ |εalias|+|εEM|+|ε ail|
wi h alias εalias = 0 when bandlimi ed + Nyquis .
4 In o ma ion Geome y and Poin e Basis
Fo mul i-channel measu emen :
Bo n p obabili y: Measu emen ou come p obabili ies pi=⟨ψ, Eiψ⟩ o POVM {Ei}.
I-p ojec ion: Minimal KL-di e gence minp∈C DKL(p∥q) o e cons ain se Cyields ex-
ponen ial amily.
Poin e basis: Windowed ope a o Ww=Rw(E)dEA(E) minimal eigensubspace (Ky
Fan) de e mines poin e basis.
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5 Discussion
Es ablished o mul i-channel sca e ing:
Phase–densi y–delay uni ica ion ia Bi man–K e˘ın–Wigne –Smi h
Channel basis co a iance and h eshold egula i y
Windowed eadou wi h NPE non-asymp o ic e o closu e
In o ma ion-geome ic Bo n p obabili y and poin e basis
Applica ions: quan um op ics mul i-mode sca e ing, mesoscopic anspo , nuclea eac-
ions, g a i a ional mul i-pola iza ion.
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